Short Answer

Evaluate \(\iiint_{\text{E}}{\sqrt{{{\text{x}}^{\text{2}}}\text{+}{{\text{y}}^{\text{2}}}}}\text{dV}\), where \(E\) is enclosed by the planes \({\rm{z = 0}}\) and \({\rm{z = x + y + 5}}\) and by the cylinders \({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ = 4}}\) and \({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ = 9}}\).

The \(\int_{\rm{0}}^{{\rm{2\pi }}} {} \int_{\rm{2}}^{\rm{3}} {\int_{\rm{0}}^{{\rm{rcos\theta + rsin\theta + 5}}} {{{\rm{r}}^{\rm{2}}}} } {\rm{cos\theta dzdrd\theta = }}\frac{{{\rm{65\pi }}}}{{\rm{4}}}\).

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Determine the integral's value.

First, swap everything into cylindrical coordinates. The area defined by these coordinates is bordered by the planes \({\rm{z = 0}}\) and \({\rm{z = rcos\theta + rsin\theta + 5}}\) is located between the two cylinders with radii of \({\rm{2}}\) and \({\rm{3}}\). The triple integral is written as:

\(\begin{aligned}\int_{\rm{0}}^{{\rm{2\pi }}} {} \int_{\rm{2}}^{\rm{3}} {\int_{\rm{0}}^{{\rm{rcos\theta + rsin\theta + 5}}} {{{\rm{r}}^{\rm{2}}}} } \rm cos\theta dzdrd\theta &= \int_{\rm{0}}^{{\rm{2\pi }}} {\int_{\rm{2}}^{\rm{3}} {{{\rm{r}}^{\rm{3}}}} } {\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{\theta + }}{{\rm{r}}^{\rm{3}}}{\rm{cos\theta sin\theta + 5}}{{\rm{r}}^{\rm{2}}}{\rm{cos\theta drd\theta }}\\\rm &= \int_{\rm{0}}^{{\rm{2\pi }}} {\frac{{{\rm{65 + 65cos2\theta }}}}{{\rm{8}}}} {\rm{ + }}\frac{{{\rm{65cos\theta sin\theta }}}}{{\rm{4}}}{\rm{ + }}\frac{{{\rm{95cos\theta }}}}{{\rm{3}}}{\rm{d\theta }}\\\rm &= \frac{{{\rm{65\pi }}}}{{\rm{4}}}\end{aligned}\)

Step 2: Result.

\(\int_{\rm{0}}^{{\rm{2\pi }}} {} \int_{\rm{2}}^{\rm{3}} {\int_{\rm{0}}^{{\rm{rcos\theta + rsin\theta + 5}}} {{{\rm{r}}^{\rm{2}}}} } {\rm{cos\theta dzdrd\theta = }}\frac{{{\rm{65\pi }}}}{{\rm{4}}}\)

Therefore, the\(\int_{\rm{0}}^{{\rm{2\pi }}} {} \int_{\rm{2}}^{\rm{3}} {\int_{\rm{0}}^{{\rm{rcos\theta + rsin\theta + 5}}} {{{\rm{r}}^{\rm{2}}}} } {\rm{cos\theta dzdrd\theta = }}\frac{{{\rm{65\pi }}}}{{\rm{4}}}\).

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Essential Calculus: Early Transcendentals

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Step 1:Determine the integral's value.

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Essential Calculus: Early Transcendentals

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